RMC, Inc. produces a variety of chemical products. In a particular production process, 3 raw materials are blended together to produce 2 materials. Each ton of fuel additive is a mixture of 2/5 ton of material A and 3/5 ton of material C. A ton of solvent base is a mixture of 1/2 ton of material A, 1/5 ton of material B, and 3/10 ton of material C. After deducting relevant costs, the profit contribution is $40 for every ton of fuel additive and $30 for every ton of solvent base produced.
RMC's production is constrained by a limited availability of the 3 raw materials. For the current production period, RMC has available 20 tons of material A, 5 tons of material B, and 21 tons of material C. The structure for this linear programming problem is as follows:
X1 = Tons of Fuel additive
X2 = Tons of Solvent base
Objective function:
Z = 40X1 + 30X2
Constraints:
Max. of Material A: .4X1 + .5X2 \leq 20
Max. of Material B: .2X2 \leq 5
Max. of Material C: .6X1 + .3X2 \leq 21
QUESTIONS:
a. Identify all extreme points of the feasible region and the associated variable values
b. Find the optima solution using the graphical solution procedure (optimal points from graph)
c. Determine the values for all slack and surplus variables and interpret each slack/surplus value
d. Use the graphical sensitivity analysis approach to determine the range of optimality for the objective function coefficients
e. Suppose RMC discovers a way to increase the profit of its solvent base to $50 per ton. Does this change the optimal solution? If so, how?
f. Use the graphical sensitivity approach to determine what happens if an additional 3 tons of material C becomes available. What is the corresponding shadow price for the constraint?
g. In addition to the original material blending constraints, suppose management adds the requirements that at least 30 tons of fuel additive and at least 15 tons of solvent base must be produced. Graph this new problem with the additional constrains and identify all extreme points for this revised problem.